A confirmation again

One of my main activities in the morning is reading the daily coming from arxiv. Sometime it happens to find significant papers to be put in a post like this. This morning I have found a beautiful paper by a cooperation of people from Germany, Russia and Australia working on lattice QCD (see here). This paper has been written by Igor Bogolubsky, Ernst-Michael Ilgenfritz, André Sternbeck and Michael Mueller-Preussker. I put here the following picture representing one of the main conclusions

propbimpsThis picture gives the gluon propagator with a number of points (96)^4 and shows clearly that it reaches a finite value at smaller momenta implying a massive gluon. Indeed, the authors of the paper extended the lattice computations moving from (80)^4 to (96)^4 points and add some other improvement in the computation itself. The value of beta is quite high being 5.7. The agreement with previous computations of Cucchieri and Mendes is excellent (see here). These latter authors worked with a number of points of (128)^4 while beta was taken to be 2.2.

The other two important conclusions they reach is that the ghost propagator goes like that of a free particle and the running coupling goes to zero at lower momenta. For the running coupling we emphasize that there is no common agreement about its definition in the infrared and the authors properly point out this. But a running coupling that goes to zero does not mean at all that there is no confinement. Quite the contrary as proved by Kazuhiko Nishijima (see here): It gives a proof of confinement.

So, we obtain again a clear proof of the scenario we have already obtained from a theoretical standpoint (see here and here) and we have discussed at length in this blog. I think that evidence of existence of the mass gap both on lattice and from theory are becoming overwhelming. We are just wating the dust to settle down and textbooks reporting these findings.

Update: After an email exchage with Andre Sternbeck he gave further clarifications about his group work correcting something not correct in the post. I post here his corrigenda:

“Our study was for the gauge group SU(3) and not for SU(2). That is
the reason why the Beta-Value is larger than that used for SU(2) by
Cucchieri et al. and by myself et al. in 2007. The lattice spacings are
roughly of the same order, but the numerical effort spent for a 96^4
lattice in SU(3) is much bigger than what had been necessary in SU(2).”

I take this chance to thank him a lot for his comments.


10 Responses to A confirmation again

  1. ervin says:


    Thank you for this informative posting. I’ve noticed that at the end of their paper the authors state the following:

    “But the lattice approach as discussed here has
    a few weak points. The choice of the gauge potentials
    Aμ(x) and correspondingly of the gauge
    functional FU(g) is by far not unique. As long
    as we are reaching the infrared limit by employing
    quite strong bare coupling values the continuum
    limit is not under control. Moreover,
    we use standard periodic boundary conditions
    which certainly have an impact on the IR limit.
    That under these conditions the gluon propagator
    does not tend to zero is related to the behavior
    of the zero-momentum modes, which do
    not become sufficiently suppressed as the lattice
    size increases. Changing the definition of
    Aμ(x), the boundary conditions and improving
    the gauge fixing procedure in order to deal properly
    with the Gribov problem may essentially
    suppress zero-momentum modes and correspondingly
    correct the behavior of both the gluon and
    ghost propagators. Therefore, a final conclusion
    still cannot be drawn.”

    Does this mean that their results are to be considered preliminary?

    Best regards,


  2. mfrasca says:

    Hi Ervin,

    The problem is due to the present understanding of Yang-Mills theory in the infrared. As you noted, in the paper, two kind of solutions are considered : One has the gluon propagator going to zero at lower momenta and the other does not. There are a lot of very smart people that worked on the first solution for a long time so that a lot of researchers believed that was just a formality to verify this. Things went the other way around and so these researchers, I think correctly, are somewhat prudent and will not consider concluded their analysis until any doubt will be dissipated on the lattice computations. This paper gives a significant improvement in this direction.

    Let me say that I appreciate a lot the work of this people that I am following since 2005 and I know of all their concerns about this matter. It should be expected some word of caution by their side.


  3. ervin says:

    Hi Marco,

    Thanks for clarifications. Here is another question, if you don’t mind. I’ve heard before talk about “off-diagonal” gluons being massive. Is this paper confirming this fact or is it completely unrelated to it?

    On a side note, I am trying to seriously learn the fascinating topic of infrared QCD (theoretical aspects, not lattice computations). With so much going on in so many directions it is very difficult to keep up with where the research is really heading. Is there an objective, reliable and up to date reference that you could suggest?



  4. mfrasca says:

    The gluon propagator is considered in the Landau gauge. In this case you will get straightforwardly

    D_{\mu\nu}^{ab}(q^2)=\delta_{ab}\left(\eta_{\mu\nu}-\frac{q_\mu q_\nu}{q^2}\right)D(q^2)

    the quantity D(q^2) in the Euclidean metric is that computed in those lattice computations. As you may know the propagator is gauge dependent.

    This is high-quality, forefront research.


  5. hermann says:

    Dear Marco.

    In a recent paper of Gong, Meng etc. their conclusion is the following:

    “The momentum space gluon propagator measured in our lattice simulations are fitted
    using various models. The exponent k, which characterize the power-law behavior of the
    gluon dressing function in the IR region, is found to be consistent with 0.5. This implies
    that the gluon propagator may have a finite value at zero momentum, in agreement with
    the result using other non-lattice methods. This work also confirms that IR region gluon
    propagator can be well investigated by adopting improved gauge action and lattices with
    large volume and coarse anisotropic spacing.”

    Do you agree to this statement?

  6. mfrasca says:

    Dear hermann,

    I have read their paper http://arxiv.org/abs/0811.4635 that is the one you are citing. The results these authors obtain are in fully agreement with the emerging scenario I am discussing in my blog and with all the lattice computations performed so far as those I have considered here in my post. The novelty in this work is in the method they use to get the results. I have not entered into the details of the lattice method they used and this should be work for referees.

    Anyhow, if their approach is correct, the instability they see in the fits of the gluon propagator is due to the fact that they are not adopting the right functional forms. Indeed, authors as the ones here in my post are not doing fitting anymore and for very good reasons.


  7. hermann says:

    In your last paper you use the LSZ reduction formula. But this method, I know, can’t be applied to bound states. Are you sure about the validity of your approach using LSZ reduction ?

  8. mfrasca says:

    Dear hermann,

    Nice question that gives me a chance to clarify an important point about my paper. If you take the argument from the start you should have noticed the particular form it takes the generating functional in the infrared. You will get a generating functional for free glueballs multiplying a generating functional for vertexes due to quark interactions with glueballs. You can introduce the pion current for the quark currents provided you will insert the pion form factor f_\pi and you will obtain the same result as for LSZ with pion free states. The particular form of the generating functional in the infrared produces the required result. Higher order corrections may hinder these conclusions but these are ineffective as showed in the theorem I give at the beginning of the paper.


  9. hermann says:

    Dear Marco.
    In your paper you put the following conjecture: “A quantum field theory can only exist with integrable classical solutions”. What kind of arguments can you give in order to give validity to this statement? I’m not sure, but a recent paper of Sergei Matinyan about quantum chaos and cosmology seems to me give arguments against this statement. Maybe I have understood his arguments not so well. What about chaotic solution of Yang Mills equation? They can’t describe any kind of field theory?

  10. mfrasca says:

    Dear hermann,

    A nice question again. Thank you. This is only a conjecture to be proved or disproved. It is based on the evidence that, so far, no QFT has ever been built using chaotic solutions and nobody knows how to eventually do this. I hope to see someone giving an answer in the near future.


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